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Design a method that can compute the following partial sum accurately for very large N, S_(N) = \ sum_(n = 1)^N ((-1)^(n-1))/(n). From Calculus, we know that \lim_(N-> \infty )S_(N) is a finite number. Since successive terms in the sum have opposite signs and become increasingly close together as n grows, there may be a danger of increasing error through cancellation. How can you evaluate the sum in a way that avoids this danger? (Hint. Use additions only.)

Design a method that can compute the following partial sum accurately for very large N, S_(N) = \ sum_(n = 1)^N ((-1)^(n-1))/(n). From Calculus, we know that \lim_(N-> \infty )S_(N) is a finite number. Since successive terms in the sum have opposite signs and become increasingly close together as n grows, there may be a danger of increasing error through cancellation. How can you evaluate the sum in a way that avoids this danger? (Hint. Use additions only.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Design a method that can compute the following partial sum accurately for very large N, S_(N) = \ sum_(n = 1)^N ((-1) ^ (n-1))/(n). From Calculus, we know that \lim_(N-> \infty )S_(N) is a finite number. Since successive terms in the sum have opposite signs and become increasingly close together as n grows, there may be a danger of increasing error through cancellation. How can you evaluate the sum in a way that avoids this danger? (Hint. Use additions only.)
Transcribed Image Text:Design a method that can compute the following partial sum accurately for very large N, S_(N) = \ sum_(n = 1)^N ((-1) ^ (n-1))/(n). From Calculus, we know that \lim_(N-> \infty )S_(N) is a finite number. Since successive terms in the sum have opposite signs and become increasingly close together as n grows, there may be a danger of increasing error through cancellation. How can you evaluate the sum in a way that avoids this danger? (Hint. Use additions only.)
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